Intuitively, the ratio \(\frac{R}{h}\)  defines the diversity of request batches in a timeslot. For example, when \(\frac{R}{h} = 1\) , requests are concentrated on a single content. When \(\frac{R}{h} = N\) , requests are spread evenly across the catalog \(\mathcal {N}\) . For that reason, we refer to \(\frac{R}{h}\)  as the diversity ratio. This definition of diversity is consistent with other notions of diversity, such as, e.g., the entropy; indeed the diversity ratio provides a lower bound on the entropy of the normalized batch vector \(\frac{\mathbf {r}_t}{R}\) , as \(E\left(\frac{\mathbf {r}_t}{R}\right) \ge \log \left(\frac{R}{h}\right)\)  [1]},
where \(E(p) = -\sum _i p_i \log (p_i)\)  is the entropy function. We note that our request model
generalizes the setting by Paschos et al.  [2]}, which can be seen as the case \(R = h = 1\)  under our request model.
We make no additional assumptions on the request arrival process; put differently, we operate in the adversarial online setting, where a potential adversary may select an arbitrary request sequence in \(\mathcal {R}_{R,h}\)  to increase system costs.
